For a subscheme V of an applicable toric variety X this command computes the push-forward of the total Segre class s(V,X) of V in X to the Chow ring of X.
i1 : setRandomSeed 72; |
i2 : R = ZZ/32749[w,y,z] o2 = R o2 : PolynomialRing |
i3 : Segre(ideal(w*y),CompMethod=>PnResidual)
2
o3 = - 4H + 2H
ZZ[H]
o3 : -----
3
H
|
i4 : A=ChowRing(R) o4 = A o4 : QuotientRing |
i5 : Segre(A,ideal(w^2*y,w*y^2))
2
o5 = - 3h + 2h
1 1
o5 : A
|
Now consider an example in \PP^2 \times \PP^2, if we input the Chow ring A the output will be returned in the same ring. To ensure proper function of the methods we build the Chow ring using the ChowRing command. We may also return a MutableHashTable.
i6 : R=MultiProjCoordRing({2,2})
o6 = R
o6 : PolynomialRing
|
i7 : r=gens R
o7 = {x , x , x , x , x , x }
0 1 2 3 4 5
o7 : List
|
i8 : A=ChowRing(R) o8 = A o8 : QuotientRing |
i9 : I=ideal(r_0^2*r_3-r_4*r_1*r_2,r_2^2*r_5)
2 2
o9 = ideal (x x - x x x , x x )
0 3 1 2 4 2 5
o9 : Ideal of R
|
i10 : Segre I
2 2 2 2 2 2
o10 = 72h h - 24h h - 12h h + 4h + 4h h + h
1 2 1 2 1 2 1 1 2 2
ZZ[h ..h ]
1 2
o10 : ----------
3 3
(h , h )
1 2
|
i11 : s1=Segre(A,I)
2 2 2 2 2 2
o11 = 72h h - 24h h - 12h h + 4h + 4h h + h
1 2 1 2 1 2 1 1 2 2
o11 : A
|
i12 : SegHash=Segre(A,I,Output=>HashForm)
o12 = MutableHashTable{...4...}
o12 : MutableHashTable
|
i13 : peek SegHash
o13 = MutableHashTable{G => 2h + h + 1 }
1 2
Glist => {1, 2h + h , 0, 0, 0}
1 2
2 2 2 2 2 2
SegreList => {0, 0, 4h + 4h h + h , - 24h h - 12h h , 72h h }
1 1 2 2 1 2 1 2 1 2
2 2 2 2 2 2
Segre => 72h h - 24h h - 12h h + 4h + 4h h + h
1 2 1 2 1 2 1 1 2 2
|
i14 : s1==SegHash#"Segre" o14 = true |
In the case where the ambient space is a toric variety which is not a product of projective spaces we must load the NormalToricVarieties package and must also input the toric variety. If the toric variety is a product of projective space it is recommended to use the form above rather than inputting the toric variety for efficiency reasons.
i15 : needsPackage "NormalToricVarieties" o15 = NormalToricVarieties o15 : Package |
i16 : Rho = {{1,0,0},{0,1,0},{0,0,1},{-1,-1,0},{0,0,-1}}
o16 = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {-1, -1, 0}, {0, 0, -1}}
o16 : List
|
i17 : Sigma = {{0,1,2},{1,2,3},{0,2,3},{0,1,4},{1,3,4},{0,3,4}}
o17 = {{0, 1, 2}, {1, 2, 3}, {0, 2, 3}, {0, 1, 4}, {1, 3, 4}, {0, 3, 4}}
o17 : List
|
i18 : X = normalToricVariety(Rho,Sigma,CoefficientRing =>ZZ/32749) o18 = X o18 : NormalToricVariety |
i19 : CheckToricVarietyValid(X) o19 = true |
i20 : R=ring(X) o20 = R o20 : PolynomialRing |
i21 : I=ideal(R_0^4*R_1,R_0*R_3*R_4*R_2-R_2^2*R_0^2)
4 2 2
o21 = ideal (x x , - x x + x x x x )
0 1 0 2 0 2 3 4
o21 : Ideal of R
|
i22 : Segre(X,I)
2 2
o22 = - 72x x + 3x + 8x x + x
3 4 3 3 4 3
ZZ[x ..x ]
0 4
o22 : -----------------------------------------
(x x , x x x , x - x , x - x , x - x )
2 4 0 1 3 0 3 1 3 2 4
|
i23 : Ch=ToricChowRing(X) o23 = Ch o23 : QuotientRing |
i24 : s3=Segre(Ch,X,I)
2 2
o24 = - 72x x + 3x + 8x x + x
3 4 3 3 4 3
o24 : Ch
|
All the examples were done using symbolic computations with Gr\"obner bases. Changing the option CompMethod to bertini will do the main computations numerically, provided Bertini is installed and configured. Note that the bertini option is only avalibe for subschemes of \PP^n.
Observe that the algorithm is a probabilistic algorithm and may give a wrong answer with a small but nonzero probability. Read more under probabilistic algorithm.
The object Segre is a method function with options.